Orthonormal basis functions for L^2(R)

In summary, the conversation discusses the existence of orthonormal bases for the space of square integrable functions over the reals, specifically focusing on the Hermite polynomials and the Haar wavelet. The Hermite polynomials are known to form an orthogonal basis, while the Haar wavelet is an example of an orthonormal basis. Further research could be done on other orthonormal bases, such as those involving wavelets with rapidly decaying functions.
  • #1
mnb96
715
5
Hello,

are there sets of functions that form an orthonormal basis for the space of square integrable functions over the reals L2(ℝ)?
According to Wikipedia the hermite polynomials form an orthogonal basis (w.r.t. to a certain weight function) for L2(ℝ). So I guess it would be a matter of multiplying the polynomials by suitable scalars in order to make them orthonormal.
Are there other known examples besides the Hermite polynomials?
 
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  • #3
Hi micromass!
thanks for your reply. Your answer basically answer my question.
Apparently the Haar wavelets "constitute a complete orthogonal system for the functions on the unit interval".

I was now wondering if there are more orthonormal bases for functions in L2(ℝ) whose support is the whole real line, e.g. rapidly decaying functions.
 

Related to Orthonormal basis functions for L^2(R)

1. What are Orthonormal Basis Functions for L^2(R)?

Orthonormal Basis Functions for L^2(R) are a set of functions that form a basis for the space of square integrable functions on the real line. They have the properties of being orthogonal to each other and having a unit norm, making them useful for representing functions in a vector space.

2. Why are Orthonormal Basis Functions important in mathematics?

Orthonormal Basis Functions are important in mathematics because they provide a way to represent functions in a vector space using a finite set of functions. This allows for efficient computation and analysis of functions, and also has applications in fields such as signal processing and quantum mechanics.

3. How do Orthonormal Basis Functions relate to Fourier series?

Orthonormal Basis Functions for L^2(R) are closely related to Fourier series, as they can be used to represent a periodic function as a sum of sines and cosines. However, Orthonormal Basis Functions can also represent non-periodic functions, and do not require the function to be square integrable.

4. Can Orthonormal Basis Functions be used for functions in higher dimensions?

Yes, Orthonormal Basis Functions can be extended to higher dimensions, such as L^2(R^n). In this case, the basis functions would be a set of n-dimensional functions that are orthogonal and have unit norm, and can be used to represent functions in n-dimensional vector spaces.

5. How are Orthonormal Basis Functions calculated?

There are various methods for calculating Orthonormal Basis Functions, depending on the specific function space and desired properties. In general, they can be obtained using techniques such as Gram-Schmidt orthogonalization or the Fourier transform. Alternatively, they may be predefined and known for certain function spaces, such as the trigonometric functions for periodic functions.

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